On invariants of surfaces in the 3-sphere

Abstract

In this paper we study isotopy classes of closed connected orientable surfaces in the standard [Formula: see text]-sphere. Such a surface splits the [Formula: see text]-sphere into two compact connected submanifolds, and by using their Heegaard splittings, we obtain a [Formula: see text]-component handlebody-link. In this paper, we first show that the equivalence class of such a 2-component handlebody-link up to attaching trivial [Formula: see text]-handles can recover the original surface. Therefore, we can reduce the study of surfaces in the [Formula: see text]-sphere to that of [Formula: see text]-component handlebody-links up to stabilizations. Then, by using [Formula: see text]-families of quandles, we construct invariants of [Formula: see text]-component handlebody-links up to attaching trivial [Formula: see text]-handles, which lead to invariants of surfaces in the [Formula: see text]-sphere. In order to see the effectiveness of our invariants, we will also show that our invariants can distinguish certain explicit surfaces in the [Formula: see text]-sphere.